A remark on p-summing norms of operators

Transcription

1 A remark o p-summig orms of operators Artem Zvavitch Abstract. I this paper we improve a result of W. B. Johso ad G. Schechtma by provig that the p-summig orm of ay operator with -dimesioal domai ca be well-approximated usig Cp log log log 2 vectors if 1 < p < 2, ad usig Cp log if 2 < p <. 1. p-summig orms Throughout this paper we will follow otatios of N. Tomczak-Jaegerma [To]. Defiitio 1.1. Let X ad Y be Baach spaces. A operator u : X Y is called p-summig if there exists a costat c such that for all fiite sequeces {x j } i X oe has j ux j p 1/p c sup x X ad x 1 j x j, x p The ifimum of costats c satisfyig this iequality is deoted by π p u ad is called the p-summig orm of u. Here we would like to discuss the followig atural questio: Give a operator u with -dimesioal domai, how may vectors do we eed to approximate the p-summig orm of u? To preset a more precise versio of this questio we will first give the followig defiitio: Defiitio 1.2. For positive iteger, let π p u is the smallest costat c such that for arbitrary vectors x 1,..., x X oe has 1/p 1/p ux j p c sup x j, x p. Oe ca see that x X ad x 1 u = π p 1 u π p 2 u π p u π p u, 2000 Mathematics Subject Classificatio. Primary 46E30, 46B07; Secodary 46B09, 60G15. Key words ad phrases. p-summig orm, L p, l l. 1 1/p.

2 2 ARTEM ZVAVITCH ad lim π p u = π p u. Now we are ready to state a precise versio of the questio above: Give a operator u with -dimesioal domai what is the smallest umber N such that π p u Cπ p N u. The first case to study is p = 2. I this case a useful result of N. Tomczak- Jaegerma see [To], page 143 states that if u is 2-summig of rak tha π 2 u 2π 2 u. C log For p = 1, S. Szarek [Sz] proved that π 1 u Cπ 1 u. Fially W. B. Johso ad G. Schechtma proved i [J-S], that π p u Cπ C log3 p u for 1 < p < 2 ad π p u Cπ C log 3 p u, for 2 < p <. Here we would like to improve a result of W. B. Johso ad G. Schechtma by reducig the power of log. We will follow a approach of [J-S] but we will use differet methods for approximatio of the expectatio or tails of a give radom process. Our goal is to prove the followig theorem: Theorem 1.1. Suppose that dimx=, u : X Y is a liear operator ad ε > 0. The π p u 1 + επ N p, as log as i: 1 < p < 2 ad N Cpε 2 log log logε 2 + log ε 1 2, ii: 2 < p < ad N Cpε 5 p 2 logε 5. Give a liear operator u : X Y of fiite rak, 1 < q < ad positive itegers, k defie ν q,k u = if where ν q u is the ifimum of where u = BwA ad { k ν q u i : u = A w B } k u i, A : X l, w : l l q diagoal, B : l q Y. The ext theorem theorem 24.2 [To] shows a coectio betwee πp u ad ν q,k u: Theorem 1.2. Let 1 p + 1 q = 1 ad the the ideal orms π p ˆν q u = lim ad ˆν q k ν,k q u, are i trace duality. The ext theorem is a key step i the proof of theorem 1.1. After provig it, we will fiish our proof of theorem 1.1 usig a iteratioal procedure, ad theorem 1.2. Theorem 1.3. Let M be positive itegers; u : X Y a liear operator with X fiite dimesioal ad dimy. The, for 1 p + 1 q = 1,

3 A REMARK ON p-summing NORMS OF OPERATORS 3 i: For 1 < p < 2, ν 7 8 M,2 q u 1 + C p M log ii: For 2 < p <, ν 7 8 M,2 q u 1 + cp M [log M ] log log M log M 2. Probabilistic lemma ν q M,1 u. ν q M,1 u. Before provig theorem 1.3 we eed to prove the followig probabilistic lemma, which will improve the result of propositio 2.1, [J-S]. The assumptios of this lemma is the Lewis lemma see [L], or [S-Z]: i [S-Z] it was show that for ay -dimesioal subspace X of L p Ω, µ, 0 < p <, oe ca fid a probability measure τ o Ω ad a subspace X of L p Ω, τ isometric to X which admits a basis h 1,..., h orthoormal i L 2 Ω, τ, such that h 2 i. I additio we ote that if X is a subspace of l M p ad τ is the probability measure o {1,..., M} give by the theorem above the, as observed i [J-S], we ca split each atom of τ of mass large tha 4/M ito pieces each of size betwee 2/M ad 4/M. This will elarge the umber of atoms by at most M/2. The ew measure λ is such that λ{i} 4/M for all i, ad it is supported o K = {1,..., K}, where K 3M/2. Fially, X is still isometric we deote this isometry by J to a subspace J X of L p K, λ, ad J X admits a orthoormal basis h 1,..., h whose sum of squares is a costat clearly, X is also isometric to J X. Lemma 2.1. Let X be a dimesioal subspace of L p K, λ with 2 < p <, ad assume that X has characteristics from the Lewis lemma plus the remark after it, cocerig the splittig of big atoms. Let B p be the uit ball of X ad {ε j } K be idepedet radom variables takig the values +1 or 1 with equal probability, the P sup λ{j}ε j xj p > cp x B p K log K 1 2. Proof: To prove this lemma we will use a method of proof due to J. Bourgai, J. Lidestrauss ad V. Milma [B-L-M].. Fix 0 < t < 1/2 to be chose later t = cp K log K Let F be a 1 2 t-et with respect of the metric dx, y = λ{j} x i p y i p o the boudary of B p the, dx, y = λ{j} x i p y i p p λ{j} max{ x i, y i } p 1 x i y i p λ{j} max{ xi, y i } p p 1/p λ{j} xi y i p 1/p cp x y p,

4 4 ARTEM ZVAVITCH so by a stadard volume argumet, we get that log F c p log t 1. It is easy to see that: P sup λ{j}ε j xj p 1 > x B p 2 t P sup λ{j}ε j xj p > t x F. Usig the Lewis lemma see [J-S], Propositio 2.1 we get that x 1/2 x p, for x X, so that the l -diameter of B p is less or equal to 2 1/2. For k = 1, 2,..., l = [log 1+t 1/2 ] + 1 let A k B p be such that B p g A k g + t1 + tk B. 3 X has characteristics from the Lewis lemma, so applyig Propositio 2.1 from [J-S] [ log NBp, B, t cp log Kt 2 ] we get: log A k cp log Kt t 2k. For every x F ad 1 k l let f k,x A k satisfy x f k,x t1 + t k /3. Put C k,x = {i : f k,x i 1 + t k 1 }, D k,x = C k,x \ l C h,x, D 0,x = {1,..., K} \ C k,x, h>k ad ˆx = xi D0,x + Note that if i C k,x, k 1, the 1 + t k I Dk,x. x i 1 + t k 1 t1 + t k /3 1 + t k 2, while if i C k,x xi 1 + t k 1 + t1 + t k /3 1 + t k+1. Hece for every i D k,x, k 1, 1 + t k 2 x i 1 + t k+2, while for i D 0,x, x i 1 + t 2. It follows that the 1 + t 2 ˆx i x i 1 + t2, 1 4pt ˆx i p x i p 1 + 4pt. The λ{j}ε j ˆxj p λ{j}ε j xj p λ{j} ˆxj p xj p c p t. So we may cosider ˆx istead of x.

5 A REMARK ON p-summing NORMS OF OPERATORS 5 To prove our lemma we first estimate the probability of the followig evet: λ{j}ε j ˆxj p I D0,x t o, x F, λ{j}ε j 1 + t pk I Dk,x t k, 1 k l, x F, where tk t. Let B k 1 k l be the collectio of all sets of form D k,x. It follows from the defiitio that log B k log A h c p log Kt t 2k. h=k Now we ca apply Berstei s iequality see, for example, [B], page 39, which states that for give {a i } R, with = 1 we get So that if the a j = a 2 i u 2 P ai ε i > u 2 exp 2 λ{j} ˆxj p j D 0,x λ 2 {j} ˆxj 2p, u = P λ{j}ε j ˆxj p I D0,x > t o 2 exp. t 0 j D 0,x λ 2 {j} ˆxj 2p, t j D 0,x λ 2 {j} ˆxj 2p Applyig λ{j} ˆxj p 2, λ{j} ˆxj p 2 K 1 + t2p, we get j D 0,x P λ{j}ε j ˆxj p I D0,x > t o 2 exp c p t 2 0K1 + t 2p. I the same maer we get: P λ{j}ε j ˆxj p I Dk,x > t k < 2 exp c p t 2 kk1 + t pk. So, i order to prove our lemma is eough to estimate 2 F exp c p t 2 0K + 2 B k exp c p t 2 kk1 + t pk = 2 exp4 log t 1 c p t 2 0K + c p expc p log Kt t 2k c p t 2 kk1 + t pk =.

6 6 ARTEM ZVAVITCH Let t o = c 1 pt, t l = c 2 pt, t k = c 2 p1 + t 2 pl k/2 t, the = 2 exp4 log t 1 c p t 2 K + c p expc p log Kt t 2k c p K1 + t 2 pl k pk t 2 = = 2 exp4 log t 1 c p t 2 K + c p expc p t t 2k log K c p K1 + t 2l pl t 5 = = 2 exp4 log t 1 c p t 2 K + c p expc p 1 + t 2k log K c p K t 5. To coclude the prove of our lemma we take t = c p log K c p K t 5 < c p. K log K i.e. such that The ext lemma shows that i some applicatios of lemma 2.1, we may omit requiremets for Lewis lemma characteristics. Lemma 2.2. Let X be a -dimesioal subspace of L p M, µ cosider JX L p K, µ, where J is the isometry defied by the splittig of big atoms. The there is a partitio K 1 K 2 of K ito two sets of cardiality at most 7 8M such that for each x JX ad j = 1, 2: i: 1 Kj x p L p K,µ C p [log M log M ] log log M x p L p M,µ, whe 1 < p < 2; ii: 1 Kj x p L p K,µ cp M log M x p L p K,µ, for 2 < p <. Proof: To prove i we use a result of M. Talagrad from his paper o embeddig -dimesioal subspaces of L p ito L N p with N ot too large see proof of Propositio 2.3 from [T] ad get that if Z L p M, τ is a isometric copy of X after a Lewis chage of desity, ad if JZ L p K, λ the 1 sup z B pjz λ{i}ε i zi p Cp K log [log K + log log K ] for most choices of sigs ε i = ±1. Oe ca see that for a fixed J the left had side is ivariat uder a chage of desity f, i.e. we ca replace the subspace Y of L p K, λ with its image uder

7 A REMARK ON p-summing NORMS OF OPERATORS 7 the atural isometry from L p K, λ to L p K, µ defied by T x = x/f 1 p. Ad usig that M < K < 3 2M we get sup µ {i}ε i zi p Cp [log M log M ] log log M x B pjx for most choices of sigs ε i = ±1. Sice also for most choices of sigs the differece betwee the umber of plus sigs ad mius sigs is less tha K/8, i follow. ii follow similarly, usig lemma Proof of the mai result Proof of Theorem 1.3: For some probability measure µ o M = {1... M}, we cosider A : Y L p M, µ, ad B : L 1 M, µ X so that A B = ν q M u ad u = Bi p,1 A, where i p,1 is the formal idetity mappig from L p M, µ oto L 1 M, µ. Next, usig lemma 2.2, we get a partitio K 1 K 2 = K ito two sets of cardiality at most 7 8M such that 1 Kj JAy p L p K,µ cp M log M Ay p L p M,µ for each y Y, j = 1, 2 ad p > 2. Deote for j = 1, 2 the ijectio from L p K j, µ K j to L 1 K j, µ K j by i j p,1 ad let P be the coditioal expectatio projectio from L 1 K, µ oto J[L 1 M, µ] followig by J 1. Thus so, usig defiitio of ν q,k u, u = B P i 1 p,1 1 K1 JA + B P i 2 p,1 1 K2 JA ν 7 8 M,2 q u 2 ν 7 8 M q [ P i j p,1 1 K j JA] 2 1 Kj JA i j p,1 BP 1 1 p 2 + cp M log M A B 1 + cp M A B 1 + cp M A B 2 µ K j 1 q 1 p log M log M. This completes the proof whe p > 2; the other case is similar.

8 8 ARTEM ZVAVITCH Fially, we are ready to prove theorem 1.1. Without loss of geerality we may assume that dimy. Usig theorem 1.2 it is eough to prove that ˆν q N v 1 + ενq M v, for all v : Y X ad all positive itegers M. Iteratig Theorem 1.3, we get that k ν [ 7 8 ]k M,2 k q u 1 + cp [ 7 8 ]j 1 M log[7 8 ]j 1 M v q M u, for all k such that 7 8 k M, ad for all p > 2. The product o the right had side of the above iequality is smaller tha 1 + ε as log as cp [ 7 8 ]j 1 M log[7 8 ]j 1 M cε. Set N = [ 7 8 ]k M, the if N Cpε 5 logε 5, the ˆν q N u ν N,2k q u ν q N u. This completes the proof whe p > 2, the case 1 < p < 2 is similar. Refereces [B] K. M. Ball A Elemetary Itroductio to Moder Covex Geometry. Flavors of geometry, 1-58, Math. Sci. Res. Ist. Publ., 31, Cambridge Uiv. Press, Cambridge, [B-L-M] J. Bourgai, J. Lidestrauss ad V. Milma, Approximatio of zooids by zootopes, Acta Math., [J-S] W.B. Johso ad G. Schechtma, Computig p-summig Norms With Few Vectors, Israel Joural of Math , [L] D. R. Lewis, Fiite dimesioal subspaces of L p, Studia Math , [S-Z] G. Schechtma, A. Zvavitch, Embeddig subspaces of L p ito l N p, 0 < p < 1, Math. Nachr pp [Sz] S. J. Szarek Computig summig orms ad type costats o few vectors, Studia Math , [T] M. Talagrad, Embeddig subspaces of L p i L N p, Geometric aspects of fuctioal aalysis Israel, , , Oper. Theory Adv. Appl., 77, Birkhuser, Basel, [To] N. Tomczak-Jaegerma, Baach-Mazur distaces ad fiite-dimesioal operator ideal, Pitma Moographs ad Surveys i Pure ad Applied Mathematics 38, Logma, Lodo, Mathematics Departmet, Mathematical Scieces Bldg, Uiversity of Missouri, Columbia, MO USA address: zvavitch@math.missouri.edu